The simplest form of \tan ^{-1} [ {\sqrt{1+x}-\sqrt{1-x}}/{\sqrt{1+x}+\sqrt{1-x}}] is

Correct Answer is : π4−12cos−1x

\frac{\pi}{4}-\frac{1}{2} \cos ^{-1} x

\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} x

CBSE Class 12 Math 2022 Term I Solved Paper - Question 31

Question : 31 of 50

Marks: +1, -0

The simplest form of

\tan ^{-1} [\;{{√{1+x}}-{√{1-x}}}/{{√{1+x}}+{√{1-x}}}]

Solution:

Explanation: We have,

\tan ^{-1} (\;{{√{1+x}}-{√{1-x}}}/{{√{1+x}}+{√{1-x}}})

Put

x=\cos 2 θ

, so that

θ=\;{1}/{2} \cos ^{-1} x

\tan ^{-1} (\;{{√{1+\cos 2 θ}}-{√{1-\cos 2 θ}}}/{{√{1+\cos 2 θ}}+{√{1-\cos 2 θ}}})

=\tan ^{-1} (\;{{√{2 \cos ^2 θ}}-{√{2 sin \;^2 θ}}}/{{√{2 \cos ^2 θ}}+{√{2 sin \;^2 θ}}})

=\tan ^{-1} (\;{\cos θ- sin \;θ}/{\cos θ+ sin \;θ})

=\tan ^{-1} (\;{1-\tan θ}/{1+\tan θ})

=\tan ^{-1}(1)-\tan -1(\tan θ)

=\tan ^{-1} (\tan ^4) -θ

=\;{π}/{4}-\;{1}/{2} \cos ^{-1} x

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